We introduce an Allen–Cahn type functional, \(\text {BE}_{\varepsilon }\) , that defines an energy on separating hypersurfaces, Y, of closed Riemannian Manifolds. We establish \(\Gamma \) -convergence of \(\text {BE}_{\varepsilon }\) to the area functional, and compute first and second variations of this functional under hypersurface perturbations. We then compute an explicit expansion for the variational formula as \(\varepsilon \rightarrow 0\) . A key component of this proof is the invertibility of the linearized Allen–Cahn equation about a solution, on the space of functions vanishing on Y. We also relate the index and nullity of \(\text {BE}_{\varepsilon }\) to the Allen–Cahn index and nullity of a corresponding solution vanishing on Y. We apply the second variation formula and index theorems to show that the family of 2p-dihedrally symmetric solutions to Allen–Cahn on \(S^1\) have index \(2p - 1\) and nullity 1.
